Spectral theory of self-adjoint operators in Hilbert space

M.S. Birman and M.Z. Solomjak

Expanding on questions traditionally treated as the core of Hilbert space theory, this book focuses on unbounded operators, develops spectral theory for a finite family of commuting self-adjoint operators and describes the unitary invariants of such families.

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  • 1. Preliminaries.- 1. Metric Spaces. Normed Spaces.- 2. Algebras and ?-Algebras of Sets.- 3. Countably Additive Functions and Measures.- 4. Measurable Functions.- 5. Integration.- 6. Function Spaces.- 2. Hilbert Space Geometry. Continuous Linear Operators.- 1. Hilbert Space. The Space L2.- 2. Orthonormal Systems.- 3. Projection Theorem. Orthogonal Expansions and Orthogonal Sums.- 4. Linear Functionals and Sesqui-linear Forms. Weak Convergence.- 5. The Algebra of Continuous Operators on H.- 6. Compact Operators.- 7. Bounded Self-adjoint Operators.- 8. Orthogonal Projections.- 9. Examples of Hilbert Spaces and Orthonormal Systems.- 10. Examples of Continuous Functionals and Operators.- 3. Unbounded Linear Operators.- 1. General Notions. Graph of an Operator.- 2. Closed Operators. Closable Operators.- 3. Adjoint Operator.- 4. Domination of Operators.- 5. Invariant Subspaces.- 6. Reducing Subspaces.- 7. Defect Number, Spectrum, and Resolvent of a Closed Operator.- 8. Skew Decompositions. Skew Reducibility.- 9. Spectral Theory of Compact Operators.- 10. Connection between the Spectral Properties of TS and ST.- 4. Symmetric and Isometric Operators.- 1. Symmetric and Self-adjoint Operators. Deficiency Indices.- 2. Isometric and Unitary Operators.- 3. Cayley Transform.- 4. Extensions of Symmetric Operators. Von Neumann's Formulae.- 5. The Operator T*T. Normal Operators.- 6. Classification of Spectral Points.- 7. Multiplication by the Independent Variable.- 8. Differentiation Operator.- 5. Spectral Measure. Integration.- 1. Basic Notions.- 2. Extension of a Spectral Measure. Product Measures.- 3. Integral with Respect to a Spectral Measure. Bounded Functions.- 4. Integral with Respect to a Spectral Measure. Unbounded Functions.- 5. An Example of Commuting Spectral Measures whose Product is not Countably Additive.- 6 Spectral Resolutions.- 1. Statements of Spectral Theorems. Functions of Operators.- 2. Spectral Theorem for Unitary Operators.- 3. Spectral Theorem for Self-adjoint Operators.- 4. Spectral Resolution of a One-parameter Unitary Group.- 5. Joint Spectral Resolution for a Finite Family of Commuting Self-adjoint Operators.- 6. Spectral Resolutions of Normal Operators.- 7 Functional Model and the Unitary Invariants of Self-adjoint Operators.- 1. Direct Integral of Hilbert Spaces.- 2. Multiplication Operators and Decomposable Operators.- 3. Generating Systems and Spectral Types.- 4. Unitary Invariants of Spectral Measure.- 5. Unitary Invariants of Self-adjoint Operators.- 6. Decomposition of a Spectral Measure into the Absolutely Continuous and the Singular Part.- 8 Some Applications of Spectral Theory.- 1. Polar Decomposition of a Closed Operator.- 2. Differential Equations of Evolution on Hilbert Space.- 3. Fourier Transform.- 4. Multiplications on L2 (Rm, Cm).- 5. Differential Operators with Constant Coefficients.- 6. Examples of Differential Operators.- 9 Perturbation Theory.- 1. Essential Spectrum. Compact Perturbations.- 2. Compact Self-adjoint and Normal Operators.- 3. Finite-dimensional Perturbations and Extensions.- 4. Continuous Perturbations.- 10 Semibounded Operators and Forms.- 1. Closed Positive Definite Forms.- 2. Semibounded Forms.- 3. Friedrichs Method of Extension of a Semibounded Operator to a Self-adjoint Operator.- 4. Fractional Powers of Operators. The Heinz Inequality.- 5. Examples of Quadratic Forms. The Sturm-Liouville Operator on [?1, 1].- 6. Examples of Quadratic Forms. One-dimensional Schrodinger Operator.- 11 Classes of Compact Operators.- 1. Canonical Representation and Singular Numbers of Compact Operators.- 2. Nuclear Operators. Trace of an Operator.- 3. Hilbert-Schmidt Operators.- 4. Sp Classes.- 5. Additional Information on Singular Numbers of Compact Operators.- 6. ?p Classes.- 7. Lidskii's Theorem.- 8. Examples of Compact Operators.- 12 Commutation Relations of Quantum Mechanics.- 1. Statement of the Problem. Auxiliary Material.- 2. Properties of (B)-systems and (C)-systems.- 3. Representations of the Bose Relations. The Case m = 1.- 4. Representations of the Bose Relations. General Case.- 5. Representations of the Canonical Relations.

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書名 Spectral theory of self-adjoint operators in Hilbert space
著作者等 Birman, M. Sh.
Solomi︠a︡k, M. Z.
Khruschev S.
Peller V.
Solomjak M. Z.
Birman M. S.
書名別名 Спектральная теория самосопряженных операторов в гильбертовом пространстве

Spektralʹnai︠a︡ teorii︠a︡ samosopri︠a︡zhennykh operatorov v gilʹbertovom prostranstve
シリーズ名 Mathematics and its applications
出版元 D. Reidel Pub. Co.
刊行年月 c1987
ページ数 xvi, 301 p.
大きさ 25 cm
ISBN 9027721793
NCID BA00525074
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言語 英語
原文言語 ロシア語
出版国 オランダ